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Let $k$ be a field, and $R=k[x,y]$.

I'm supposed to find two different minimal primary decompositions of the ideal $(x^2y, y^2x)$.

It's easy to see that one minimal primary decomposition is $(x)\cap(y)\cap(x^2, y^2)$.

My question: What's the second minimal primary decomposition of the above ideal?

EDIT: I've now added the minimality requirement to the question.

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Is it necessary that they be minimal? –  Qiaochu Yuan May 23 '11 at 19:06
    
For some reason I can't edit my original question, so here is an important change (which answers Qiaochu's question): Yes, the decompositions have to be minimal. –  user11281 May 23 '11 at 19:32
    
Your accounts have been merged. Since your new account is registered you should no longer have problems with logging in. –  Qiaochu Yuan May 23 '11 at 19:53

2 Answers 2

Since you suggest this is homework, I'll put this in the form of hints:

(1) Remember that the set of associated primes for a primary decomposition is unique. So you are looking for $\mathfrak{p} \cap \mathfrak{q} \cap \mathfrak{r}$ where $\sqrt{\mathfrak{p}} = (x)$, $\sqrt{\mathfrak{p}} = (y)$ and $\sqrt{\mathfrak{r}} = (x,y)$.

(2) Show that $\mathfrak{p}$ and $\mathfrak{q}$ must be $(x)$ and $(y)$. So the place where you have room to play is in choosing $\mathfrak{r}$.

At this point, I have trouble giving good general hints. The next two, which are far more helpful, will be put in ROT13.

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(4) Gur rknzcyr V sbhaq jnf nyfb trarengrq va qrterr gjb.

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I'm a bit late with this but even so:

$$ \begin{align} (x^2 y, y^2 x) &= (x^2 y - y^2 x, y^2 x) \\ &= (xy(x-y), y^2 x) \\ &= (y) \cap (x(x-y), y^2x) \\ &= (y) \cap (x) \cap (x-y, y^2) \end{align}$$

It remains to be shown that $(x-y, y^2)$ is primary:

Let $r(I)$ denote the radical of $I$. Then $r((x-y, y^2)) = (x-y, y) = (x,y)$. To see that $(x,y)$ is maximal note that $k[x,y]/(x,y) \cong k$ is a field. We know that if $r(I)$ is maximal then $I$ is primary so we're done.

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